Group #11782609 — subgroups

Subgroups — #11782609

Order 1152.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10593388	C₂	1	4	yes	maximal
2	#10590905	C₂	1	2	yes	maximal
3	C₈×#622	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×Dic₁₂	—	2	2	no	—
5	#63873	C₂²	1	4	yes	—
6	C₂×#3718	—	2	2	no	—
7	C₄×#622	C₂²	1	2	yes	—
8	#63743	—	2	8	no	—
9	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
10	#5183	—	6	2	no	—
11	C₂×C₃×Dic₆	—	2	4	no	—
12	C₂×#622	D₄	1	4	yes	—
13	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
14	C₄×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
16	C₃×Dic₁₂	—	4	4	no	—
17	#3724	—	2	8	no	—
18	#3718	—	4	4	no	—
19	#1713	—	9	1	no	maximal, Sylow
20	C₂×C₃×Q₁₆	—	6	2	no	—
21	#944	—	6	4	no	—
22	C₂×#216	—	6	4	no	—
23	C₂×Dic₁₂	—	2	2	no	—
24	C₈×Dic₃	—	6	2	no	—
25	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
26	C₂×C₃⋊Dic₃	C₂×D₄	1	2	yes	—
27	C₂×C₃×Dic₃	—	4	4	no	—
28	C₃×Dic₆	—	4	4	no	—
29	C₃×Dic₆	—	4	4	no	—
30	#622	Q₁₆	1	8	yes	complemented
31	C₃²×C₈	D₈	1	2	yes	—
32	C₃⋊(C₃⋊C₈)	D₈	1	2	yes	—
33	#500	—	9	4	no	—
34	#494	—	9	2	no	—
35	C₈²	—	9	1	no	—
36	C₂×C₃×Q₈	—	6	4	no	—
37	C₂×Dic₆	—	2	4	no	—
38	C₃×Q₁₆	—	12	4	no	—
39	C₂×C₃×C₈	—	2	2	no	—
40	#216	—	12	8	no	—
41	Dic₃⋊C₄	—	12	8	no	—
42	C₄×Dic₃	—	6	2	no	—
43	C₂×C₃⋊C₈	—	6	2	no	—
44	Dic₁₂	—	4	4	no	—
45	C₂²×C₃²	#127	1	1	yes	characteristic
46	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
47	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
48	C₃⋊Dic₃	C₂×Q₁₆	1	4	yes	—
49	C₃×Dic₃	—	8	4	no	—
50	C₂×Q₁₆	—	6	2	no	—
51	C₂×Q₁₆	—	18	2	no	—
52	C₄⋊Q₈	—	9	4	no	—
53	#107	—	18	8	no	—
54	C₄×C₈	—	9	2	no	—
55	C₄×C₈	—	9	1	no	—
56	C₃×Q₈	—	12	4	no	—
57	C₃×Q₈	—	12	4	no	—
58	C₂×C₃×C₄	—	2	2	no	—
59	C₂×C₃×C₄	—	12	4	no	—
60	C₂×Dic₃	—	4	4	no	—
61	C₂×Dic₃	—	6	4	no	—
62	Dic₆	—	4	4	no	—
63	Dic₆	—	4	4	no	—
64	C₃×C₈	—	2	4	no	—
65	C₃⋊C₈	—	6	4	no	—
66	C₂×C₃²	#493	1	1	yes	characteristic
67	C₂×C₃²	#494	1	2	yes	—
68	C₂×Q₈	—	6	4	no	—
69	Q₁₆	—	12	4	no	—
70	Q₁₆	—	36	4	no	—
71	C₂×C₈	—	9	4	no	—
72	C₂×C₈	—	9	1	no	—
73	C₂×C₈	GO⁺(4,𝔽₂)	1	1	yes	characteristic
74	C₄⋊C₄	—	18	8	no	—
75	C₄²	—	9	1	no	—
76	C₂²×C₃	—	2	2	no	—
77	C₃×C₄	—	2	2	no	—
78	C₃×C₄	—	2	2	no	—
79	C₃×C₄	—	24	4	no	—
80	Dic₃	—	6	8	no	—
81	Dic₃	—	8	4	no	—
82	C₃²	#1713	1	1	yes	characteristic, complemented, Sylow
83	Q₈	—	12	4	no	—
84	Q₈	—	12	4	no	—
85	C₂×C₄	—	9	2	no	—
86	C₂×C₄	—	12	4	no	—
87	C₂×C₄	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
88	C₈	—	9	8	no	—
89	C₈	—	9	2	no	—
90	C₈	#3774	1	2	yes	—
91	C₂×C₃	—	2	4	no	—
92	C₂×C₃	—	2	2	no	—
93	C₂²	#64231	1	1	yes	characteristic
94	C₄	—	9	4	no	—
95	C₄	—	24	4	no	—
96	C₄	#64225	1	1	yes	characteristic
97	C₄	C₂×#3774	1	1	yes	characteristic
98	C₃	—	2	2	no	—
99	C₂	#10593295	1	2	yes	—
100	C₂	#10593418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593388

C₂

yes

maximal

#10590905

C₂

yes

maximal

C₈×#622

C₂

yes

maximal, characteristic

C₂×C₃×Dic₁₂

—

#63873

C₂²

yes

—

C₂×#3718

—

C₄×#622

C₂²

yes

—

#63743

—

C₈×C₃⋊Dic₃

C₂²

yes

characteristic

#5183

—

C₂×C₃×Dic₆

—

C₂×#622

D₄

yes

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

C₃×Dic₁₂

—

—

—

—

C₂×C₃×Q₁₆

—

#944

—

C₂×#216

—

C₂×Dic₁₂

—

C₈×Dic₃

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×D₄

yes

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

#622

Q₁₆

yes

complemented

C₃²×C₈

D₈

yes

—

C₃⋊(C₃⋊C₈)

D₈

yes

—

#500

—

#494

—

C₈²

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

#216

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

C₂×D₈

yes

characteristic

C₃²×C₄

C₂×D₈

yes

characteristic

C₃⋊Dic₃

C₂×Q₁₆

yes

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#107

—

C₄×C₈

—

C₄×C₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#493

yes

characteristic

C₂×C₃²

#494

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

GO⁺(4,𝔽₂)

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#1713

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×GO⁺(4,𝔽₂)

yes

characteristic

C₈

—

C₈

—

C₈

#3774

yes

—

C₂×C₃

—

C₂×C₃

—

C₂²

#64231

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₃

—

C₂

#10593295

yes

—

100

C₂

#10593418

yes

characteristic

	subgroup	G/H
center	C₂²	#64231
commutator, cosocle	C₄×C₃⋊Dic₃	C₂³
Frattini	C₂×C₄	C₂×GO⁺(4,𝔽₂)
Fitting	C₂×C₃²×C₈	D₄
socle, prime radical	C₂²×C₃²	#127
hypercenter	C₂×C₈	GO⁺(4,𝔽₂)
hypocenter	C₃²	#1713
LCS	G ▷ C₄×C₃⋊Dic₃ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

#64231

commutator, cosocle

C₄×C₃⋊Dic₃

C₂³

Frattini

C₂×C₄

C₂×GO⁺(4,𝔽₂)

Fitting

C₂×C₃²×C₈

D₄

socle, prime radical

C₂²×C₃²

#127

hypercenter

C₂×C₈

GO⁺(4,𝔽₂)

hypocenter

C₃²

#1713

LCS

G ▷ C₄×C₃⋊Dic₃ ▷ C₂²×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
30. #622	Q₁₆	semidirect
82. C₃²	#1713	semidirect

H ≅ G/N

type

30. #622

Q₁₆

semidirect

82. C₃²

#1713

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10593388	2
#10590905	2
C₈×#622	2
#1713	9

subgroup

index

Subgroups — #11782609

Conjugacy classes of subgroups

Complements

Maximal subgroups